But pv = ⅔T/m
Hence p being constant,
pdv = ⅔ dT/m
Therefore σ′ = σ + ⅔ 1/m
Now suppose an amount of heat, dH, is given to a single molecule and that its temperature is T. Its specific heat is σ, and
dH = σmdT
But dH = βdT
Therefore β = σm
Hence 1/m = σ/β
Thus σ′ = σ(1 + 2/(3β))
And σ′/σ = γ
Therefore γ = 1 + 2/(3β)
Or β = 2/(3(γ-1))